Question

Can someone explain to me the difference between linear, first -order; linear, second-order; non-linear, first order and non-linear, second order.

i need to formally define them.

Answer 1

So a differential equation is of nth order if the higher order derivative in the equation is nth order.

E.g. y'' - 17/y' + y^2 = sin x
is 2nd order, because y'' is the highest order derivative

Now the basic way to think about linearity is this: an nth order equation is linear if it is linear in each of the derivatives: linear in y, linear in y', ..., linear in y^(n)

Th equation above is clearly not linear because it is not linear in either y' or y, although it is linear in y''

Answer 2

But seriously; this is why you have a text book. These terms should be explained there.

Answer 3

The first five minutes of this video will help you as well:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-3-solving-first-order-linear-odes/

and you might want to go hunting among other lecture titles for other examples.

Answer 4

i have three text books and the answer isnt in any of them... its been a while since i took Calc or DE and i am familiar with the terms, i just need to formally tell the difference between the 4.